How to move vertices into normal angles
You can create a Custom Orientation from the edge you want to square to. That's the little '+' in the header's Orientations dropdown. If you have to do this a lot, I would recommend creating a shortcut for it, and/or adding it to your Quick Favourites menu.
(You can also set Custom Orientations to be used after creation, and to overwrite previous creations, to avoid clutter, in the F9 'Adjust Last Operation' panel.)
'Y' then becomes the direction along the edge. If you set the Pivot point to 'Active Element', it's quite quick to select the vertex to be moved, then the vertex to square to, and SY0. (Scale the vertices to 0, along Y, to the active vertex.)

However, the result is only one of the possible edges in the plane at right-angles to the target edge.
You can use the same strategy to flatten vertices onto a plane of your choice. Select any 3 vertices to define the plane's direction, and again, create a Custom Orientation from them. Select any set of vertices, with  a vertex lying in the the plane you want to flatten to active, and this time, SZ0.